Update 4.37pm: The Independent Alliance is seeking a joint meeting with acting Taoiseach Enda Kenny and Fianna Fáil leader Micheál Martin and themselves "in the next 24 hours" in a bid to force a breakthrough to the political stalemate, write Fiachra Ó Cionnaith and Juno McEnroe.
Independent Alliance members Michael Fitzmaurice, Finian McGrath and John Halligan confirmed the plan to the Irish Examiner as the group's six TDs meet to discuss the situation facing the country, which later saw a statement released detailing the move.
Under the plan, the Shane Ross-led group wants the two party leaders to sit down with it "as a matter of urgency" as it is "anxious to resolve the current deadlock" and wants to see a real "dialogue" between Mr Kenny and Mr Martin.
It also believes exact details are required "from both parties" over the "scope and timetable any minority government may be allowed to govern within".
"We believe both parties should be able to reach agreement on mutual support for at least three budgets, so as to provide the country with the stability which all parties have acknowledged is necessary.
"These issues need to be clarified immediately," the Alliance's statement read.
It is understood Fine Gael and Fianna Fáil have yet to make any decision on the proposal.
However, it is far from certain it will take place with relations between both Mr Kenny and Martin taking a severe hit this week over the failed "equal partnership" talks.
The Independent Alliance proposal has been made after a day of increasing pressure on Independents to make a decision on which party - if any - they will back in next Thursday's latest taoiseach nomination vote in the Dáil.
Update 1.31pm: Fianna Fáil’s Jack Chambers has claimed a coalition with Fine Gael “will not happen”, writes Elaine Loughlin.
He added that it is now “up to the Independents to decide” on the type of minority government that will be formed.
“The Independents will have to decide which party they will go into a minority government they will go into,” he told RTÉ’s News at One.
“We want to see real Dáil reform. I don’t believe a grand coalition between Fine Gael, Fianna Fáil and the Independents will offer that reform.”
Update 1pm: Fianna Fáil TD Timmy Dooley has attempted to put the pressure back on Independent TDs, saying they can end the political stalemate next Thursday and that "it's there's to call", writes Fiachra Ó Cionnaith, Political Reporter.
The party's transport spokesperson made the comment after his colleague Eamon O Cuiv earlier said any party that gets close to 60 seats in the next Taoiseach nomination vote may be able to form a minority government.
Speaking on Newstalk's Lunchtime programme, Clare TD Timmy Dooley said what happens next "will depend on how the Independents decide to move over the coming days".
While claiming he does not "want to put pressure on them", he said "it's there's to call".
Mr Dooley denied his party is "shirking responsibility" and insisted he and other Fianna Fáil TDs do not want to be forced to "reflect" on breaking pre-election promises not to enter power with Fine Gael when the next election takes place.
He said Micheál Martin believed when he sat down to speak with acting Taoiseach Enda Kenny on Wednesday evening that only a minority government scenario was on offer, and that this remains an option for Fianna Fáil.
Fianna Fáil has made it clear it is still focussed on leading a minority government and is seeking to organise talks for Monday or Tuesday next week, with phone calls expected long before those days.
However, a number of Independents - who are meeting with themselves at 5pm this evening - are believed to be frustrated by the larger parties’ positions and may not back either on Thursday.
Asked when a government may be formed, Mr Dooley said he would "hope" by the end of this month.
However, he noted this is the Grand National horse racing weekend and that "you may want to lay that bet off somewhere else as well".
Update 12 noon: President Michael D Higgins will not intervene in the political stand-off unless he is directly requested to do so by caretaker Taoiseach Enda Kenny, writes political reporter Fiachra Ó Cionnaith.
A spokesperson for President Higgins said the country's most senior figure has no plans to intervene in the crisis.
He said there has been no contact between President Higgins and Mr Kenny, and that Áras an Uachtaráin has no view either way on whether one should be forthcoming.
Under articles 13 and 16 Constitution, the sitting president may only intervene in political matters under certain circumstances that are deliberately restrictive in order to ensure the independence of the Dáil and Seanad.
Update 11.45am: Waterford Independent TD John Halligan has said it is unlikely the Independents will support either Enda Kenny or Micheal Martin for Taoiseach next week in the Dáil, writes political editor Daniel McConnell.
Mr Halligan was scathing in his criticism of both Mr Kenny and Mr Martin in not being able to do a deal.
Speaking to the Irish Examiner, he said: “It is appalling, it is outrageous. We are livid and sick. We have taken a lot of flack from our own supporters about possibly doing a deal. But we stood up to the plate. But we will not be used as cannon fodder by either of them."
Asked if he could see himself supporting either man in the Dáil for the vote next week, Mr Halligan replied in the negative.
“No, I can’t. I have to speak to the rest of the lads but at this stage no. And given what they have done, to hell with them both. Their behaviour is pathetic and childlike.
"When you think of the wars going on around the world and we can’t get two democratically elected leaders to sort it out. It is despicable,” Mr Halligan added.
Update 10.30am: Kerry TD Michael Healy Rae (pictured above) has said he is disappointed to have put "two weeks into talks and FF put 10 minutes into talks with Fine Gael" about the formation of a government.
He added: "No one will thank you for causing a second election. We'd have the same result and be back in same position again."
Some have suggested that President Michael D Higgins (pictured below) could take a hand in negotiations if no outcome can be reached - he has the power to refuse to dissolve the Dáil or address the Oireachtas.
Constitutional law expert Dr Conor O'Mahony said the rules governing such an intervention make it problematic.
He explained: "There is a power for the President to address the Houses of the Oireachtas. But the President can only do so on the consent of the Government, so the content of the address has to be agreed and cleared.
"The Government in this case is Fine Gael and Labour, so the perception could arise there is some degree of favouring one political party over another."
Fine Gael has "done everything it possibly could" to form a government in recent weeks, and remains open to talks with Fianna Fáil on forming a partnership government, acting Health Minister Leo Varadkar has said.
Speaking on RTE's Morning Ireland, he said: "It's absolutely the case that Fine Gael is still open to talks with Fianna Fáil."
He suggested the best way forward was for the two parties to meet on policy issues first, and look at forming a shared policy agenda, before talking specifically about a structure that would support that agenda.
However, he said Fine Gael did not believe a Fianna Fáil-led minority government was viable.
"It's a simple matter of arithmetic really," he said, as Fianna Fáil has only 43 seats in the 158-seat Dáil.
"You might as well talk about a Sinn Féin-led minority government, or one led by one of the smaller parties. It's just not viable," he said.
Mr Varadkar said the Fine Gael offer of a partnership government still stood.
"That's for three reasons," he said. "First is that it's in the best interests of the country, second that it respects the outcome of the election and third it would provide us with a stable and lasting government to help heal our fragmented society."
He said while he had not expected Fianna Fáil to accept the proposal for partnership immediately, he was surprised at the haste with which they rejected it.
"Fine Gael has done everything it possibly could in the last month to form a government," he said, referring to talks with Independents, smaller parties and culminating in the offer to Fianna Fáil.
"We're keen to get a Government in place for the country."